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8 We give a fully dynamic (Las-Vegas style) algorithm with constant expected amortized time per 9 update that maintains a proper (∆ + 1)-vertex coloring of a graph with maximum degree at most ∆. 10 This improves upon the previous O(log ∆)-time algorithm by Bhattacharya et al. (SODA 2018). 11 Our algorithm uses an approach based on assigning random ranks to vertices and does not need to 12 maintain a hierarchical graph decomposition. We show that our result does not only have optimal 13 running time, but is also optimal in the sense that already deciding whether a ∆-coloring exists in a 14 dynamically changing graph with maximum degree at most ∆ takes Ω(logn) time per operation. 15 2012 ACM Subject Classification Theory of computation → Dynamic graph algorithms 16

Constant-Time Dynamic (Δ+1)-Coloring.